Interaction between solitons in a stable medium
V. E. Zakharov and A. B. Shabat
Novosibirsk State University
(Submitted December 18, 19721.
Zh. Eksp. Tear. Fiz 64,1627-1639 (May 1973)
The nonlinear equation (1) is solved exactly by the inverse scattering problem method.
Interaction between solitons is studied within the framework of the equation, the soliton
shifts due to their collisions with each other are calculated, and it is shown that only
paired collisions occur. The results are applied to the problem of diffraction in a nonlinear defocusing medium.
I NTRODUCTI ON
The equation
iUt
+ U xx -
?{
Ju J'u =
(1)
0
is encountered in various physical problems. Gross [1J
and Pitaevskil' [2J, and later on Tsuzuki [3J, used this
equation to describe the oscillations of a Bose gas at
zero temperature. The problem of the evolution of the
complex envelope of a monochromatic wave in a nonlinear medium [4,5J also leads to Eq. (1). Finally, interpreting t as a longitudinal coordinate, Eq. (1) can be
treated as a two-dimensional variant of the well known
"parabolic" equation[6J that describes stationary wave
beams in a nonlinear medium.
Equation (1) belongs to a class of equations that can
be solved exactly by the method of the inverse scattering
problem. This was demonstrated by the present authors[8 J in a detailed study of the case K < 0 corresponding, when dealing with a Bose gas, to attraction of its
particles. In this case the trivial spatially-homogeneous
solution of Eq. (1) of the type u = Aexp (-iK IAI 2t), which
we shall call a condensate in analogy with the case of
the Bose gas, is unstable. In the nonlinear-optics interpretation of Eq. (1) the condensate has a meaning of a
stationary monochromatic wave, and the instability of
the condensate at K < 0 corresponds to self-modulation
instability of a monochromatic wave [ 7J.
In the present paper we study the case K > 0 corresponding to repulsion of Bose-gas particles and to stability of a monochromatic wave relative to self-modulation. In the "spatial" interpretation of the parameter t,
the case K > 0 corresponds to the propagation of a wave
beam in a defocusing medium. As will be shown in the
present paper, reversal of the sign of K leads not only
to a change in the physical picture of the phenomena
described by Eq. (1), but also to a considerable restructuring of the mathematical formalism necessary for its
solution.
When the method of the inverse scattering problem is
used[9,lOJ, the initial nonlinear equation is associated
with a linear differential operator L, which in our case
takes the form (see [7J)
. [ 1 + PO] -+
a [ 0 u' ]
L=,
o 1 - p ax
u O'
2
p' - 1 .
?{=--
(2)
The sought function u is a coefficient of this operator.
.T:! can be shown that the scattering matrix of the operator
tion u(± 00) = O. Such a formulation is natural for the
case of attraction. Of greater physical interest for the
stable case is lu(x, t) 12 - const, llx - 0 as x - ± 00,
corresponding to the propagation of waves through a
condensate of constant density. In such a condensate the
soliton
V+U(X,t)
()" + iv)' + exp {2v(x - Xo - -:--~,.
2)"t)}
-'-_.-:----;-:-'--c:---'---____
v =
1 [- cxp {2v (x - Xo - 2)"t)}
1'1 - ),,'
can move with constant velocity. In this case
~
?{
T1u(x, t) I' = 1
ch'v(x _
Xo _
2)"t)
The parameter A characterizes the amplitude and velocity of the soliton, and Xo is the position of its center at
t = O.
Solitons play the same role in Eq. (1) as in the
Korteveg-de Vries (KdV) equation [9,11, 12J. They are
directly connected with the discrete spectrum of the
operator L, namely, each soliton corresponds to a timeindependent eigenvalue of the operator. Just as in the
KdV case, the solitons determine the asymptotic behavior
of an arbitrary initial condition as t - co and behave in
simple fashion when scattered by each other.
In Secs. 1 and 2 of the present paper we consider the
qirect and inverse scattering problems for the operator
L at K > O. The condition lu(± 00)1 2 = const calls for the
study of the analyticity of the scattering matrix on a twosheet Riemann surface, and this makes the theory of the
inverse problem much more complicated in comparison
with the case u(± 00) = O. The results of Secs. 1 and 2
are used in Sec. 3 to study soliton collisions. The scattering of solitons is considered and the shifts of the centers of the solitons after their collisions are calculated.
Unlike Tsuzuki [3 J , we calculate not only the relative
values of the shifts, which follow from the conservation
law for the mass center, but also their absolute magnitudes. We prove also the additivity of the shifts in collisions of a large number of solitons. In Sec. 4 we consider the problem of reflection of a soliton from a wall
on which the wave function of the condensate vanishes.
In Sec. 5 we apply the results to the problem of the
diffraction of a plane wave by an opaque band in a nonlinear defocusing medium.
1. THE DIRECT SCATTERING PROBLEM
L varies in time in accordance with a very Simple law,
and the solution of (1) reduces to a reconstruction of the
"potential" u(x, t) from a given scattering matrix (the
solution of the inverse scatteringyroblem). This problem was solved for the operator L in [8J under the condi-
We consider the problem of the eigenvalues of the
operator f.:
823
Copyright © 1974 American Institute of Physics
Sov. Phys.-JETP. Vol. 37, No.5, November 1973
(3)
ix =
EX.
X = [~:l.
We make the change of variables
823
Xt = (p -1)'/' exp { i p' ~ 1
X
X' = (p
}Vt,
+ 1)'/' exp{ i p'~ 1X }v,.
(4)
form as x - - 00 into the asymptotic form as x _ + 00,
In the notation of (6) and (7) we have
The system LX'" E X is reduced by it to the form
aV
t
i - + q'v,
ax
[ ::: ] =
pE
1.=--
AVt,
=
p'-l '
a~
ax + qVt = AV"
q=
- i-
(5)
u
(p' -1) 'I,
x + = e-
i ,.
[
I
X + = e't. [
1]
~ -A"
S11 = S'" = a,
A]
l'
Here t(A) = (A2 _1)112 is a double-valued function of A.
Analogously, as x - - 00 we have
X - = e- itx [
t
J
1
e'"(~-A)I"
X- =
eit~
(7)
[e-'O(~-A)]
l '
1/1,-+-x,+
as
x-+-+
Qlt = al/1t + bl/1"
+ b'l/1l.
oo •
Analogously
We denote the Wronskian of two functions by {u, v}
= UIV2 - U2Vl' From (13) we have
a (A ~) = {<pt, 1/1,} = _
,
II"" 1/1,1
{<PI, 1/1,1
- A) .
00
Sljt (x, s)X + (s, A) ds.
(8)
t
{<pt, <p,} = det S{l/1t, 1/1,}.
Since {<pI. <p2}
= {l/!l, l/!2} =-2 t (t
- A), it follows that
( ~+~) ['¥11(X,y)
ax
( a
ay
] =
a )['¥,,(x,y)]
'I",,(x,y)
a;-iiii
i[
'I",,(x,y)
J..y (x, s)X,+ (s, A)ds,
.[
-1
'I",,(xy)
it follows that this function can be analytically continued
to the upper sheet of the Riemann surface, and has as
1A 1 - 00 the asymptotic form
The functions l/! 1 and l/!2 are analytic on the lower sheet
of the Riemann surface.
(9a)
It follows from (14) that a(A, ?;) is analytic on the
upper sheet of the Riemann surface. The zeros of a
correspond to the eigenvalues of the system (5). From
the self-adjoint character of the system and from relation (15) it follows that they lie on the segment -1 < A
< 1 of the real axis and are simple. From (14), (17),
and (18) it follows that as IAI - 00 the coefficient a(A, t)
behaves asymptotically like
1 -q'(x) ]['¥11(X,y)]
-1
'¥,,(x y) ,
q(x)
2'¥'t2(X, x) =2'¥,,(x, x) =i(q(x) -1);
as
y -+-
+
00.
It can be shown (cf. [13J) that the system (9) with these
boundary conditions can be uniquely solved, from which
follows the existence of the representation (8). The system (9) has a symmetry, by virtue of which
(10)
The system (5) is invariant with respect to the involution
(11)
in particular, X 2 = Xl and l/!2 = If l' This circumstance,
as well as the relations (10), enable us to find a triangular repr~sentation for l/!2. It is effected with the same
matrix w( x, y) as used for l/! 1.
a(A,~) "" [(~-A)'e'o-ll /2~(~-A).
Sov. Phys..JETP, Vol. 37, No.5, November 1973
(19)
Of particular interest is the case b(A) == O. Then the
coefficient a(A, t) is determined completely by its zeros
Ak (k = 1, ... , N) and is a meromorphic function on the
complete Riemann surface. Indeed, in this case we have
jal 2 = 1 on the cuts and a(A) can be continued to the lower
sheet in accordance with the symmetry principle
a (A', ~') =1Ja'(A, ~).
(20)
In the lower sheet, at the points Ai' the coefficient a has
simple poles, This circumstance and the condition (19)
at infinity enable us to determine the form of a(A, t):
The scattering problem for the system (5) consists of
constructing a matrix S that transforms the asymptotic
824
(17)
Analogous properties are possessed by the Jost function
<Pl(X, A). Its asymptotic form is
<Pt(x,A) -+-Xt-(x,A)[1+0(1IjAj)l.
:(18)
(9)
'
with boundary conditions
'¥j,(x, y) -+- 0
(16)
1 - q'(x) ] ['¥,,(x,y)]
q(x)
=!
(15)
So far we defined the Jost functions for real A and t.
The function t(A) is defined on a two-sheet Riemann
surface with cuts (- 00, -1) and (1, 00), On the upper
sheet of the surface we have 1m?; > 0 and on the lower
1m t < O. From the triangular representation for the
function l/!2
1/1,(x, A) -+-X,+(x, 1.)[1+0(1/ jAlll.
Substituting (8) i,.n (5) we verify, after transformations,
that the matrix w(x, y) should satisfy the following system of partial differential equations
(14)
2~ (~
If u and v are the solutions of the system (5), then their
Wronskian does not depend on x. From (13) we obtain
1/12 (x, A) = X,+ (x, A) -
We assume that the Jost function l/! 1 can be expressed in
the triangular representation
I/1t (x, A) = Xt+ (x, A) -
(13)
(13a)
'1', = a'l/1,
detS=jaj'-jbj'=1.
We determine, for real t, the Jost functions l/! 1 and l/!2
as solutions of the system (5) with asymptotic form
I/1t-+-Xt+,
S" = S'" = b.
The asymptotic relation (12) can be rewritten in terms
of relations between the Jost functions
(6)
~-
(12)
From the invariance to involution, it follows that
.
We assume that Iql - 1 as x - ± 00. Generally speaking,
in this case q - eiO'j;.as x - ± 00. However, t,he change
of variables q - qe-10'+, VI - VI, V2 - v2e-10'-t: makes
it possible to put q - 1 as x - + 00. Then q - e 10' as
x - - 00. The quantity 0' = 0'_ - 0'+, as will be shown below, does not depend on the time. The asymptotic solution of the system (5) as x - 00 is given by
s [::~].
(21)
V. E. Zakharov and A. B. Shabat
824
From a comparison of (19) and (21) it follows that the
total change of the phase of the condensate q (x) is determined by the distribution of the eigenvalues
S
In a (I., ~) =
[11>' (x, I.) - 11>' (-
00,
I.) ldx,
from which we get
+~
(22)
The use of the operator £. for the solution of Eq. (1)
is based on the identity of this equation with the operator
relation
at I at = i[i, A),
(23)
f n = SUn(X,I.)-!n(-oo,A)ldx.
All the quantities In are conservation laws of (1).
We present in explicit form the first field In:
+~
f.=-S {lql'-1}dx,
where (see [8J)
~
A
=
lOa'
p
ax'
[0 1]
-
+
lui'
.. ]
+ p lUx
. lui' .
1
[
-
p-1
lUx
From (23) we find that the eigenfunctions X of the
operator f. satisfy the equation
(24)
which goes over as Ixl - 00 into an equation with constant
coefficients. In (24) we obtain for v 1 at large Ix I
1) u,-2il. ax,
au
au, (I.'
ifit= p+ 1+p
a'u,
-p ax' '
from which it follows that
a (1., t)
=
b(I., t)
a (1., 0) =const,
b(I., O)e-""".
=
It is seen from (14) that A = Ai the functions cp 1 and
l/!2 are proportional to each other
=
b,l/l, (x,
A,).
In those cases when b(A) can be analytically continued
in the vicinity of the point Ai' we have b i = b(Ai)' This
enables us to find the dependence of bi on the time:
b,(t)
b,(O)exp{41.;v,t}.
=
dq I '
(26)
We consider the asymptotic form of the quantity
In a (1.,
~) '"
f.
~
l:
(d
'
d'q .•
+ d;"lql') +ldx,I-2}dx.
The first three quantities II, 12, and Is have, apart from
the numerical coefficients, the meaning of the conservation laws for the particle number, for the momentum,
and for the energy of the Bose gas. The remaining conservation laws have no Simple physical meaning.
2. THE INVERSE SCATTERING PROBLEM
Formulas (25) and (26) enable us to calculate the
scattering matrix at an arbitrary instant of time from
the initial data. This yields the quantity u(x, t) at an
arbitrary instant of time, provided that the inverse
scattering problem is solved for the system (5). In the
present section we construct integral equations
(Marchenko equations) which make it possible to determine the potential q(x) from the scattering data.
We start from relation (13), which we rewrite in the
form
The independence of the coefficient a of the time
shows that Eq. (1) has a continual set of conservation
laws. It is possible to distinguish among them a denumerable set of conservation laws that are expressed in
explicit form in terms of q(x).
.
"ITxI
(25)
The eigenvalues Ai' as the zeros of a(A, 1;), do not depend on the time.
<jJ,(x, I.,)
+~
f'=-1{2I q "+6I q
(2il.)'·
It=!
as I AI - 00 and at Re A > O. The coefficients Ik can be
found directly from the system (5). To this end we make
the change of variables
-
1 (1-CPt-X\+ ) .
2,,~
e'~11
a
=
(28)
(I/l'-X,+
b
) e"'-,
. 1
+-I/l,
a
2,,~
y>x.
The left-hand side of (28) is analytic on the upper sheet
of the Riemann surface, with the exception of the points
An' at which it has simple poles. As IA I - 00, the lefthand side of (28) behaves like exp{-lm 1;(Y -x)}O(1/IAi).
We integrate relation (28) along the contour indicated
in the figure. In the integration of the left-hand side,
the contour can be closed through infinity, so that the
integral of the left-hand side is equal to the sum of the
residues in An:
From the system (5) we get an equation for <1>:
d 1
dx q'
q '~-Ill'
From (27) we
+ 1ll"-lql'=2iAIll'.
easil~ obtain an asymptotic serie~ for
~ !.(x)
11> (x, 1.) = .t...J (2il.)' '
,
where the quantities fk(x) are determined by the recurrence formula
/;f.,
!,=-Iql'.
Hk=n-1
It is obvious that
825
ImA
..- .-----
<1>':
I
I
k=l
!n=q'~dd
~!n-'+ .t...J
"\"1
xq
(27)
Sov. Phys.-JETP, Vol. 37, No.5, November 1973
/
"-
\
\
\
\
---ry-x-"+",,,~;,::;:::::::;::::::;.=i\::;:::::= ReA
\
i
\
/
--- -,
/
I
\
, '-
'
........
-
V. E. Zakharov and A. B. Shabat
/
I
/
/
--'/
825
_1_
2n
J~~ (~!p,
_ X, +)
a
eit'dA =
~
£..l
'P: (x, A~)
v"a (An,
F,(z) = -IlAe-"', F,(z) = -Ile-",
= ~ b n "P~ (x, An) e-"'.
£..l vna (A", iv,,)
Solving the system (33), we get
(29)
'l' (x )=_ v(A+iv)
IV n )
The integral of the right-hand side can be transformed
into
Jee ("P,-x,+ +,~",,)
a
it
_1_
2n
•
(1)
='ir(x,y) [O]+[-F,
1
)[-F.
•
(1)'
(x+ y )]
(x+Y)+lF,
F,(x+ y)
(I)
DO
_JA(X
• 'l' ,s
dA
.
(30)
+ 2M)
(36)
3. INTERACTION OF SOLITONS
1
!p, (A, e), "', (A, -~) = e _ A "', (A, e).
Substituting these formulas in (13) and (13a), we readily
obtain
b(A, -e) = b'(A, C)ei . ,
whence
c",(-e) = c",'(e).
(31)
It follows from (31) that the functions Fi~)2(Z) are reaL
Using the triangular representation for 1{!2 in (29), we
observe that (29) reduces to the form (30) if Fi~)2(Z) is
replaced by Fi~~(z), where
Fi 2J,(z)=- L,llnAnexP(-VnZ),
We note that in accordance with (26) and (32) we have
dlnp./dt = 4AV. Introducing
We denote the Single-soliton solution (35) by
qo(x - 2At, A, xo). In the more general case when b(A)
== 0 and there are N eigenvalues, the system (33) determines a 2N-parameter particular solution of Eq. (1)
which can be expressed in explicit form. Just as in[8,uJ,
we shall call this an N-soliton solution.
c = b / a.
C(A, -e) = C'(A, e),
(35)
and comparing (35) with (2), we verify that the eigenvalue
A corresponds to a soliton moving with velocity 2A.
Unlike the KdV case, the velocity can be of arbitrary
sign.
ds.
From the system (5) (see (13a)) we get directly
a(A, -e) = a'(A, e)e'",
e*-u),
(x)=1-2i'l' '(x x)= (A+iv)'+(vlll)e''"
q
12,
1 + (vlll)e'"
In Il = 2v (x,
(s+.y)]
(I)
1 +~
,
F, (z)=-J c,(~)e-"'ds,
2n
e'·
1 + (vlll)e""
,y
= (1- A') 'h.
(1)'
(s+Y)+lF,
F,(s+ y)
Here
CPt (A, - e) = ~ _ f"
12
v
F,(')(z)=- Ellnexp(-V"z),
Iln:= bn/v"a'(A n, iv n).
Let us consider the problem of the interaction of two
solitons with velocities 2Al and 2A2. To this end it is
necessary to study the corresponding two-soliton solution. Owing to the complexity of the explicit formula for
this solution, we confine ourselves to an investigation of
its asymptotic behavior as t - ± 00.
Proceeding as in [llJ, we verify that as t - ± 00 the
two-soliton solution breaks up into individual solitons:
q(x, t) ~ q,(x - 2A,t, A" x,+) + q,(x - 2A2t, I." x,+), t ~ +00,
q(x, t) ~ q,(x - 2/"t, A" x,-) + q,(x - 2A,t, A" X2-), t ~ -00.
The scattering of the solitons by each other gives rise to
shifts of their centers
(32)
Combining both parts of (28), we obtain a system of two
integral equations (Marchenko equations) for the quantities Wu and WIll, which determine the triangular representation of the Jost functions I{! 1 and 1{!2:
.
Let us calculate these shifts. We note first that, as seen
from (35), each soliton with velocity 2Ai represents a
discontinuity in the phase of the wave function of the condensate
argq(-oo) -argq(+oo) =2arctg(v,f1.;) =cz, .
'l',,(x, y)- J{'l'I2(X, s)F,(s + y) + 'l' .. (x, s) [- F, (s + y) + iF,' (s + y)]} ds
= F, (x
'l'1I(x,y)= -
+ y) -
iF,' (x
+ y),
J{'l'II(x,s)F,(s+ y)
The total phase discontinuity in the presence of N eigenvalues is
(33)
It is interesting to note that it does not depend on the
+ 'l',,(x, s) [- F, (s
Here
F, (z) = F,O) (z)
+ y)- iF,' (s+
+Fi') (z),
relative positions of the solitons.
y)]} ds = - F,(x+ y).
Returning to the case of two solitons, we consider the
Jost function CPl(X, A). As x - -00, it takes the asymptotic form
F, (z) = Fi" (z) + F;') (z).
From (25) and (26) we can obtain the time dependences
of the functions F 1 ,2. It is easy to verify that these func-
tions satisfy a system of linear requations
at,
aF,
a'F,
aF,
aFt
Tt+4Tz=4~, Ot-+4o;-=O.
(34)
The system (33) contains the solution of the inverse
scattering problem for the operator L.
In the particular case when b(A) == 0, Fi 1 )2 == 0, the
integral equations are degenerate, and the system (33)
reduces to a system with a finite number of linear algebraic equations (2N if N is the number of eigenvalues).
In the simplest case of a Single eigenvalue A, we have
826
Sov. Phys.·JETP, Vol. 37, No.5, November 1973
CPt (x,A) -+- e'" [ exp (ira,
+1a ,)} (v _
-1 < A < 1.
A)]'
The asymptotic form of cP I(X, A) as x - + 00 is
'P,(x, 1.)-+ a(A,iv)e'" [ w-'}..
. 1 ],
CPt (x, A) ~
A""A, , A, ,
b", (t) r'" [ iv", ~ 1.,,2 ],
A = A" A,.
We represent the coefficient a in the form a
(see (21)), where
iv
+A-
iv
+ A + iv", + A",
V. E. Zakharov and A.B. Shabat
(37)
= ala2
iv", - A,,2
826
is the component of the "single-soliton" scattering matrix.
4. REFLECTION OF A SOLITON FROM A
BOUNDARY
Let A1 > A2. As t - - 00, the solitons move apart to
large distances and Iql - 1 in the region between solitons. As t - -00, soliton 2 (with velocity 2A2) is located
on the right, so that q - exp (iQl2) in the region between
the solitons. As t - 00, soliton 1 is located on the right,
and q - exp (iQl1) in the region between the solitons.
The entire preceding reasoning pertained to the case
of an infinite condensate. It can be extended, however, to
include the case of a semi-infinite (0 < x < 00) condensate with a reflecting boundary on which zero boundary
conditions are specified. We note for this purpose that
the soliton solutions of (1) include a singular "standing"
SOliton, for which A = 0 and v = 1. The standing solitons,
with its center at the origin, is given by
We consider in this region the asymptotic form of the
Jost functions cp 1(X, A) as t - - 00. We have
(jl, ""
a, (I., iv) e" [
. (lV-')..
I
) ], I.
el(x~
e''''(iv'-A')]
(jl,""b,-(t)e-'" [
1
q(x)
* I."
1.=1.,.
Here b1(t) is a function as yet unknown. To calculate it
we note that by virtue of (20) we have a(A, -iv)
= 1/a"'(A, iv). "Passing" the function CP1 through soliton
2 in accordance with this rule, and comparing with (37),
we obtain
b,-(t) =a,(A" iv,)b,(t)
In the same manner we determine the quantity bji(t)
= b2(t)/a1(A2, i V2)'
The quantities b1(t) and b 2(t) determine the positions
of the solitons as t - - 00 (see formulas (32) and (36)).
We introduce analogously the quantities b:i(t) and b;(t),
we determine the positions of the solitons as t - 00.
Reasoning as before, we obtain
b,+(t) = b,(t) / a,(A" iv,),
b,'(t) = b2 (t)a,'(A" iv,).
From (32) and (36) we have for the soliton shifts
1
b,+
1
_1_ln (1.,-1..)' +(v, + v,)'
6x, =-In-=-In~~..,.......,~
2v,
b,2v,
la,(A" iv,) I'
2v, (A,-A,)'+(v,-v,)"
(38)
1
.,
1 (A,-A,)'+(v,+v,)'
1 b,+
6x,=-ln-=-lnla'(A' 'v,) I =--In":--~---'--~
2v, b,- 2v,
'
2v, (A,-A,)'+(v,-v,)"
Thus, the soliton 1, which has the larger velocity, acquires a positive shift, and soliton 2 a negative shift.
Solitons behave like repelling one-dimensional particles.
From (38) we get the relation
v,6x, + v,6x,
=
O.
(39)
This relation was obtained by Tsuzuki [3J directly from
Eq, (1) by analyzing the motion of the mass center of a
Bose gas. It can be interpreted as a condition for the
conservation of the mass center of the solitons during
the collision.
We proceed now to the case of N solitons. Reasoning
just as in [l1J, we can show that as t - ± 00 an arbitrary
N-soliton solution breaks up into individual solitons.
The sequence of the solitons as the prime changes from
- 00 to + 00 is reversed, so that each soliton collides with
each other soliton. Repeating the argument given above,
we verify that the total shift of a soliton, regardless of
the detailed picture of the colliSions, is equal to the sum
of the shifts in individual collisions:
<,),=x/-x,-= ~o.;,
The equations (33) of the inverse problem yield an
antisymmetrical potential q(x) if the function Fi1)(z) is
odd and the function F~ 1) (z) is even. In addition, there
should exist an eigenvalue A = 0, J.l = 1. The remaining
discrete spectrum should be symmetrical, namely, to
each An there should correspond A_n = -An and J.l- n
= 1/J.l n ·
This reasoning enables us to calculate the shift of the
soliton as it is reflected from the boundary. Such a reflection can be regarded as a double scattering, viz., by
a soliton with opposite velocity and by a standing soliton.
For a soliton with velOCity 2 A, calculating with the aid of
formulas (40), we obtain
1
1
6,,=slgn(A'-A;)-2
In
,
Vi
(I., - I.,)' + (v, + v;)'
(I.i - 1.)'+(
)'
j
Vi-Vj
1
6=-ln--.
v 1-v
The shift of the soliton when it is reflected from the
boundary is pOSitive-the soliton is repelled by the
boundary.
5, DIFFRACTION BY A BAND IN A NONLINEAR
DEFOCUSING MEDIUM
The problem of Fraunhofer diffraction of a plane
monochromatic wave by a band in a nonlinear defOCUSing
medium leads to Eq. (1) with initial conditions
ul,_o=o at Ixl <a,
ul,_o=1 at Ixl >a.
(42)
Here t and x are the longitudinal and transverse coordinates, and a is the half-width of the band; the amplitude
of the infinite wave is set equal to unity.
In accordance with the results of Sec. 1, it is necessary to solve the eigenvalue problem (5) with the function q = ul t =0' We have at Ixl < a
1] e-'" + c [0]
1 e'" '
v = cI [ 0
2
(43)
at x >a
v=c,[ i(1-A')"'-A
1
]exp {-(1-A')'''x},
(40)
..
(41)
Since the wave function of the condensate vanishes at the
reflecting boundary, the standing soliton describes
(at x > 0) the state of the condensate in the presence of
a boundary. Formula (41) continues it in antisymmetrical fashion into the region x < O. Such a continuation
can be effected for any solution of Eq. (1) satisfying the
zero condition on the plane boundary.
and finally, at x
j+i
=thx.
V
(44)
< -a
_
[
- c,.(
1
0)"
, 1-1."-1.
]cxp
. {( 1 - I.)
, 'I, x ) .
(45)
An analogous fact was established earlier for the KdV
equation[l1,12] and for Eq. (1) atx < 0[8J.
Making the solution (43)-(45) continuous at the point
x = ± a, we obtain after elementary transformations the
eigenvalue equation
827
V. E. Zakharov and A. B. Shabat
Sov. Phys.·JETP, Vol. 37, No.5, November 1973
827
cos 2Aa = A.
(46)
Equation (46) has a set of symmetrically disposed zeros
±A n, and, as expected. IAnl < 1.
At sufficiently small a, there is only one pair of
zeros. At large a, the number of pairs of zeros N can
be estimated from the formula N ~ 2a/w.
The zeros of Eq. (46) correspond to solitons that have
in the given case the meaning of partial-shadow bands
propagating along straight lines in the (x, t) plane. The
soliton with eigenvalue An makes an angle 'Pn = tan- 12A n
with the wave direction. The minimum number of such
bands is equal to two.
Thus, the diffraction of a plane wave by a band in a
nonlinear medium differs in principle from the diffraction in a linear medium in that it can be observed at an
arbitrarily large distance from the band, whereas in a
linear medium the diffraction picture becomes "smeared
out" at large distances.
CONCLUSION
Soliton solutions are only particular solutions of Eq.
(1), inasmuch as the general case b(A) f. 0, and the practical reconstruction of q(x, t) from the scattering data is
a very complicated problem. It can be assumed, however, that the N -soliton solution "contained" in any
initial condition determines its asymptotic behavior
as t - 00. This conclusion is based on the fact that in
the linear approximation there can propagate in the condensate waves whose minimal group velocity (speed of
sound) coincides with the maximum velocity of the
soliton. This circumstance leads to a separation of the
soliton in the non-soliton parts of the solution, with the
non-soliton part going outside the limit of the "sound
cone" Ixl ~ 21tl, in which the solitons are contained, and
its amplitude decreases like t-1 / 2 , owing to the dispersion spreading.
828
Sov. Phys.·JETP, Vol. 37, No.5, November 1973
The entire preceding theory was one -dimensional and
in this sense strongly idealized. One can hope, however, that it will be useful also in the solution of nonone-dimensional problems, at least such in which the
dependence on the transverse coordinates is weak. This
pertains primarily to the problem of the stability of a
soliton with respect to long transverse perturbations.
E. P. Gorss, Nuovo Cimento 20, 454, 1961.
L. P. Pitaevskil, Zh. Eksp. Teor. Fiz. 40, 646 (1961)
[Sov. Phys.-JETP 13, 451 (1961)].
3 T• Tsuzuki, J. Low Temp. Phys. 4,441,1971.
4y. E. Zakharov, Zh. Eksp. Teor. Fiz. 53, 1733 (1967)
[SOV. Phys.-JETP 26, 994 (1968)] •
5 y . I. Karpman, Nelinelnye volny v dispergiruyushchikh
sredakh (Nonlinear Waves in Dispersive Media),
Novosibirsk state Univ., 1968.
6 y . I. Talanov, ZhETF Pis. Red. 2, 223 (1965) [JETP
Lett. 2, 141 (1965)].
7y. I. Talanov and y. N. Bespalov, ZhETF Pis. Red. 3,
471 (1966) [JETP Lett. 3, 307 (1966)].
By. E. Zakharov and A. B. Shabat, Zh. Eksp. Teor. Fiz.
61, 118 (1971) [SOV. Phys.-JETP 34, 62 (1972)].
9 C. S. Gardner, G. Green, M. Kruskal and R. Miura,
Phys. Rev. Lett. 19, 1095, 1967.
lOp. D. Lax, Comm. on Pure and Appl. Math., 21, 467,
1968.
lly. E. Zakharov, Zh. Eksp. Teor. Fiz. 60, 993 (1971)
[SOV. Phys.-JETP 33, 538 (1971)] •
12 A. B. Shabat, Dinamika sploshnol sredy (Dynamics of
a Continuous Medium), No.5, 130 (1970).
13 Z. S. Agranovich and Y. A. Marchenko, Obratnaya
zadacha kvantovol teorii rasseyaniya (The Inverse
Problem of the Quantum Theory of Scattering),
Khar'kov, 1959.
1
2
Translated by J. G. Adashko
177
V. E. Zakharov and A.B. Shabat
828